A boundary integral equation approach to computing eigenvalues of the Stokes operator

• Published in: Advances in computational mathematics Vol. 46; no. 2
• Main Authors: Askham, Travis, Rachh, Manas
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Boundary integral method; Computational fluid dynamics; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Computer simulation; Divergence; Domains; Eigenvalues; Eigenvectors; Integral equations; Mathematical analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operators (mathematics); Robustness (mathematics); Visualization; Wave propagation; 45B05; 34L16; Eigenvalue; Integral equation; 76D07; Stokes flow; Fredholm determinant
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-020-09774-2

The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier–Stokes equations. As the Stokes operator is second order and has the divergence-free constraint, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high-frequency “pollution” when using Green’s function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply- and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.